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Continued Fractions

Extracts from this document...

Introduction


Continued Fractions

A continued fraction is any mathematical expression in the form of:

image00.png

image01.png

Where a0 is always and integer, and all other ‘image10.png’s such as a1, a2, and a3 are positive integers.  The number of terms can either be finite or infinite.  A more convenient way to denote continued fractions such as the one above would be to denote it by:     image19.png

Finite Continued Fractions

A finite continued fraction is an expression such as the one shown above which could end.  Every rational number can be equated to a finite continued fraction.  The only skill needed would be division of fractions.

image23.png

image25.png

Infinite Continued Fractions

Unlike the finite continued fractions, the chain of fractions never ends in an infinite continued fraction.  Every irrational number can be equated to an infinite continued fraction.  This fact was discovered and proven by the Swiss Mathematician, Leonhard Euler (1707-1783).  Some of Euler’s infinite continued fractions are as we will see below:

image31.png

A way to summarise this expression is to let image37.pngdenote the value of the continued fraction.

image43.pngimage38.png

Usage of Continued Fractions

Continued fractions could be used to solve certain quadratic equations of the second degree.  Solving a quadratic equation using the

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Middle

 will continuously fluctuate but start to stabilize when image03.png=8.  Once image03.png reaches 23, image20.png will converge to a sustained value of 1.618033989.  This value will remain constant as long as image03.pngimage21.png23

The same trend is brought up by the graph of image03.png and image22.png.  As the value of image03.png rises, image22.png will oscillate until image03.png reaches 25.  Only then will the value of  image22.png converge to a consistent value of 0.  This value will stay the same as long as image03.pngimage21.png25.

  1.  When trying to determine the 200th term, we couldn’t obtain a negative value, only a positive value.  When the value of  image03.png is still below 25, it oscillates between values which sometimes range from negative to positive values, but then as the value of image03.png increases it will converge to a constant specific positive value.  That means a negative value could only be obtained if  image24.png
  1. According to the table in question two, the exact value for the continued fraction is  1.
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Conclusion

k were to be a negative number, no value could be obtained because a math error occurs and k must be an integer because that is the rule of a continued fraction;  if it weren’t an integer, though values could still be obtained unlike a negative k, it will no longer be considered a continued fraction.

Bibliography

http://images.google.com.sg/imgres?imgurl=http://upload.wikimedia.org/math/a/b/d/abda6e157fb26b5fb84d43df36a8ad31.png&imgrefurl=http://en.wikipedia.org/wiki/Solving_quadratic_equations_with_continued_fractions&h=169&w=296&sz=2&hl=en&start=14&um=1&usg=__bg1UGcYe_dOsasg8Zdzt-Vawfjk=&tbnid=qfG0CexZoK9QDM:&tbnh=66&tbnw=116&prev=/images%3Fq%3Dcontinued%2Binfinite%2Bfraction%26um%3D1%26hl%3Den%26sa%3DG

http://images.google.com.sg/imgres?imgurl=http://www.jcu.edu/math/vignettes/v18im7.gif&imgrefurl=http://www.jcu.edu/math/vignettes/continued.htm&h=279&w=322&sz=3&hl=en&start=9&um=1&usg=__2U0RUn8JPqKD-Mom_NAD3Wge-M4=&tbnid=agACwiBwpt7vbM:&tbnh=102&tbnw=118&prev=/images%3Fq%3Dcontinued%2Bfinite%2Bfractions%26um%3D1%26hl%3Den%26sa%3DG

END OF PORTFOLIO

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